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Full text of "On the Curvature of Space"

\3 Biodiversity 
^Heritage 
^^Library 

http://www.biodiversitylibrary.org/ 



Proceedings of the section of sciences / Koninklijke Akademie van 

Wetenschappen te Amsterdam. 

Amsterdam Johannes Muller,[1 899-1 937]. 
http://www.biodiversitylibrary.Org/bibliography/1 401 7 



v.20 pt.2 1918: http://www.biodiversitylibrary.org/item/49194 
Page(s): Page 1 309, Page 1 310, Page 1311, Page 1 31 2, Page 1 313 



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Mechanics. 



- — "Further remarks on the solutions of the field- 
equations of Einstein's theory of gravitation' . By Prof. W. 
dk Sitter. 



(Communicated in the meeting of April 26, 1917). 

1, Einstein has recently ; ) enounced the postulate, that a solution 
of the field-equations 

Gu» — £ A <jy, =s — n Tjn -f 4 k g rJ T ..... (1) 



vv 



in order to be admissible for the actual physical world, must have 
no discontinuities "at finite distances". In particular the determinant 
g must for all points at finite distances be different from zero. 
This postulate is not fulfilled by my solution B, as Einstein very 
correctly points out, and as is also shown very clearly in my 
communications. This postulate, however, in the form in which it 
is enounced by Einstein, is a philosophical, or metaphysical, postulate. 
To make it a physical one, the words: "all points at finite 
distances" must be replaced by "all physically accessible points". 
And if Ihe postulate is thus formulated, my solution B does fulfil it. 
For the discontinuity arises for 

r — r l = ^ jx R. 

This is at a finite distance in space, but it is physically in- 
accessible, as I have already pointed out a ). The time needed by a 
ray of light, and a fortiori by a moving material point, to travel 
from any point r, ip t to a point v xt tp lt #, (tp a and i9-, being 
arbitrary) is infinite. The singularity at r = r x can thus never affect 
any physical experiment, or as I expressed it I.e., the paradoxical 
phenomena, or rather absence of phenomena, resulting from this 
singularity, can only happen before the beginning, or after the 
end of eternity. 

2. A similar remark lias been made by Prof. Felix Klein, in 
a letter to the present writer \\AieA J 91 s April 19. He writes: 



l ) Kritisches zu einer von Herryi dr Sitter gegebenen Liming der Gravitations- 
gletchungm , Sitzungsber. Berlin, 7 March 1918, page 270. 

s ) On Einstein's theory of gravitation and its astronomical consequences, third 
paper, Monthly Notices of the R. A. S. Vol. LXXVII1, page 17—18. 

On the curvature of space, these Proceedings, Vol XX, p. 229. 

90 
Proceedings Royal Acad. Amsterdam. Vol. XX. 



1310 

"Denken Sie sicli die gauze vierdimensionale Welt von Welt- 
linien darehfureht. Nun seheint es doch bei alien Ansatzen im 
BiNSTEiN'sclien Siune eine notwendige physikalische Voraussetzung 
zu sein, dass man diese Linien, so wie sie sicli kontinuierlich an 
einander reilien, mit einem positiveu Richtungssinn versehen kann (der 
von der "Vergangenheit" zur "Zukunft" fiilut). Dies ist nun im 
Falle B nicht moglieh. Lege ich nam 1 ich eiuer ersten Linie nach 
Belieben einen positiveu Sinn bei und iibertrage diesen unter 
Beach tung der Kontinuitat auf die Nachbarlinien, so komme ich 
sehliesslich, wegen der Zusamnaenhangsverhaltnisse des Elliptisclien 
Ratlines, zur Ausgangslinie mit umgekehrtem Sinn zur tick. Es ent- 
spricht das dem Umstande, dasz die Ebene der elliptisclien (wie der 
projekti vise hen) Geometrie eine einseitige Flaehe ist, bei der sieh die 
Indicatrix (J, die ich um irgend einen Punkt der Ebene herum 
legen mag, wenn ich sie liings einer durch den Punkt laufenden 
Geraden versehiehe, bei Ruckkehr zum Ausgangspunkte umgekehrt 
hat: O* Meine Bemerkung in Math. Annalen 37, p. 557 — 58: 
dass die Uebertragung der Schering'schen Potentialtheorie auf den Fall 
der elliptisclien Ebene unstatthaft ist, ruht genau auf demselben 
Umstande". 

Prof. Klein's remark is undoubtedy correct: we return to the 
starting point with the positive direction reversed, but only if we 
have travelled along a straight line, or at least along a line which 
intersects the polar line of the starting point. This "motion", though 
mathematically thinkable, is physically impossible, for the same 
reason as above. If we travel along an arbitrary closed curve, 
which does not intersect the polar line of any of its points, i.e. 
if we describe a physically passible circuit, then we shall, on 
returning to the starting point, find the positive direction unaltered. 

In my former paper J ) I pointed out that, in spherical space, the 
potential # 44 — 1 becomes infinite at the antipodal point. I concluded 
therefrom that, for the representation of the actual physical world, 
the elliptical space is to be preferred to the spherical. Prof. Klein 
has already made the same remark in his paper of 1890, quoted 
at the end of his letter. He points out, however, that in elliptical 
space the sign of the potential would be ambiguous. This would 
be the case if the above mentioned circuit were possible. Since it 
is impossible we can choose one of the two possible signs without 
the danger that any physical phenomena or experiments will ever 
lead to contradiction or indeterminateness. 



l ) On the curvature of space, these Proceedings, Vol XX, p. 240. 



1311 



I use this occasion to point out that, as is well known, Prof. 
Klein was the first to call attention to the elliptical space and its 
relation to and difference from the spherical space, and generally 
to investigate and explain the different possibilities of non-Euclidean 
geometry a ). In fact all geometrical concepts used in the different 
stages of the development of modern physical theory are contained 
in Klkin's general scheme as given in the second of the papers 
quoted in the footnote, 

3. If we start from the assumption, that the gravitational field is 
of such a nature that it is possible, by introducing a suitable system 
of coordinates, to bring the line-element into the form 

<jfc" = — a dr* — b (d^ + sin 7 ip thV) -f fdt\ . . . (2) 

then we can call r the "radius- vector" and t the "time". If now 
we add the condition that a % />, f must be functions of r only, and 
not of t, \p t (>, then these conditions may be briefly expressed by 
saying that the field is static and isotropic. Then the line-element of 
three-dimensional space is 



dif = a dr* -f b [dtp 2 -f «V \p diV] 



(3) 



and consequently we have 

ds* 



do* -\- fdf 1 . 



(2') 



If now we add the hypothesis that do 7 shall be the line-element 
of a space of constant curvature, thus 

r= R . x 

do % = R* j </v* -f sin* x [<*■!>' -4- «V ty rf* f ] }» 



(3') 



then the field-equations (J J reduce to one equation for f, of which 
the solutions A and B are 

/ = c Q cos 1 % ...... . {±B) 

If we drop the condition of isotropy, then f may be a function 
of r, ip, i>. For this case Levi-Civita s ) has given the general solu- 
tion of the differential equation for /. He starts from Einstbin's 
original equation, i.e. the equation (1) with I = 0. It is however 
not difficult to extend the proof to the general case. Then the 
equation (11') of Levi Civita [\. C. page 530] replaces (11) [p. 526 J, 



l ) Ueber die sogenannie Nicht-Muelidische Geometrie, Math. Annalen, Band 4 
and 6 (1871 and 1872). 

Programm mini Eintritl in die philo&ophische Facilitate Evlamgen 1872, reprint- 
ed Math, Annalen, Band 43, p, 63* 

3 ) ReoMa fisica di alcuni spazi normali del Blanch i, Rendiconti della R. 
Accad. dei Lincei t Vol. XXVI, p, 519 (May 1917). 

90* 



1312 

and consequently we must nse K -j- zp — X instead of Ii -\- xp. 
The equation (13) is not affected, but we now have K* = K — 
| (3 K -j- itp — 2), and again R* = K; therefore instead of (14) we 
find 3 K + yq) — >l = 0, but the general solution of (13) remains the 
same. This solution, expressed in my notation, is 

|//^ a 9 cos x + a l sin % sin /?, ....*. (5) 

ft being the " latitude" referred to a plane of symmetry, whose 
inclination f and node # on the plane t[> = are given by 

b x — a x !i sin £ sin & af 



b 



a 



• ' 



6 a = — «j 7t! s?"« t roa i^,, 



b % = a x R cos e* 

b . . , h l being the constants of integration introduced by Levi-Civita. 
The condition of isotropy now is a x = 0. If this is introduced 
(5) is reduced to (4 B), and it thus appears thai my solution B is 
the general solution for the case of a static and isotropic gravitational 
field in the absence of matter. 






Meteorology. — "The Atmospherical Circulation above Australasia 
according to the P Hatha I loon- Observations made at Batavia*** 
By W. van Bemmklkn. (Communicated by Di\ J, P. van der Stok), 

l Communicated in the meeting of April 26, 191 8-) 



During the years 1909 — '17 numerous determinations of direction 
and velocity of the wind at different altitudes were made by means 
of pilotbal loon-observations, by the personnel of the Observatory of 
Batavia; in the beginning with a view to investigating the general 
system of aircurrents above the West of Java, later on aiming at 
special purposes. Just for this reason the series of observations is in 
no way homogeneous. Elsewhere 1 will treat this heterogeneousness 
in extenso, here however briefly. At present the general results about 
direction and velocity of the principal aircurrents in the different 
levels will be given. 

In recent years a more or less detailed report on these observa- 
tions has sometimes been given l ) ; for this occasion the whole material 
has been subjected to a comprehensive calculation, first the observa- 
tions have come to an end by lack of balloons, secondly as it was 
considered advisable not to wait any longer with such a compre- 
hensive survey. 

A few observations about the endeavours to render the series 
more homogeneous may be given here. In the first place for levels 
under 3 km. observations made between t> a.m. and 9 a.m. have 
been taken into account only, in order to limit as much as possible 
the disturbing influence of land- and sea-breezes, which reach about 
to the 3 km. level and are weak during those morning hours. 
Moreover to the results for those lowest levels a special correction 
derived from the results obtained for the diurnal and semidiurnal 
variation of the wind has been applied 3 ). 

In the second place of balloon-observations made outside Batavia 



l ) Zitdngsverslag van 25 Juni 1910 (These Proa Vol XVIII, p. 149) 

Observatorium, Batavia; Verhandelingen N°, L 

Nature Vol. 87, 1911; Vol. 90, 1912; Vol. 91, 1914. 

Luftfahrt und Wissenschaft, Berlin (J. Sticker) Heft 5, 1913. 

Natuurkundig Tijdschrift, Dl. 73, 1913. 
«) Zittingaverslag van 26 Mei 1917, (These Proc, Vol XX, p. 119).